Question

Find the G.P. if $T_5:T_{10}=32:1$ and $T_7=\frac{1}{32}$

Answer 1

Given : $T_5:T_{10}=32:1$

$⟹\frac{T_5}{T_{10}}=\frac{32}{1}$

$⟹\frac{a{r}^4}{a{r}^9}=\frac{32}{1}$

$⟹r^5=\frac{1}{32}$

$⟹r=\frac{1}{2}$

Also, $T_7=\frac{1}{32}$ ........ [Given]

$⟹a{r}^6=\frac{1}{32}$

$⟹a{\left(\frac{1}{2}\right)}^6=\frac{1}{32}$

$⟹a\times \frac{1}{64}=\frac{1}{32}$

$⟹a=2$

Thus, the G.P. is given by $2,2\times \frac{1}{2},2\times \frac{1}{2^2},2\times \frac{1}{2^3},..........$

Hence, the G.P. is $2,1,\frac{1}{2},\frac{1}{2^2},.........$